Answers will vary. Many people will notice that the table is symmetrical; if you look at any two points directly across the main diagonal (from top left to bottom right), you will see that the values are equal. Other patterns include the fact that the same number appears on each diagonal from bottom left to top right (one diagonal has nothing but 9s, for example), and the fact that each number appears in every row and column exactly once.

b.

The diagonal pattern exists because these are all numbers that add up to the same value in the real number system. Moving one unit up reduces the sum by 1, and moving one unit to the right increases the sum by 1, so moving along this type of diagonal (up and right, or down and left) does not change the sum of the two numbers.

For the number 4, we are looking for a number x so that 4 + x = 0 in the system. We want a value so that 0 will be the units digit of the sum. If the sum is 10, then 0 will be the units digit. Therefore, x is 6, and 4 + 6 = 0 in the system.

b.

Yes, every number in the table has an additive inverse. Here is a table:

Yes, this law holds. The main reason is that the law is true in real numbers, so it must also be true in this system (which is based directly on real number arithmetic). You can also see this from the table; (a + b) and (b + a) are opposite each other on the main diagonal.

Finding the answer to 7 - 3 is the same as finding a number x so that 3 + x = 7 in the system. To find this number, we can look at Row 3 in our table to see all the possible results we can get from 3 + x. In this case, the result in Column 4 gives 7, so 3 + 4 = 7, and 4 (the column value) is the solution to 7 - 3.

As a more complicated example, let's find 2 - 9. This is the same as finding a number x so that 9 + x = 2 in the system. Looking in the row for 9, we want to find a result of 2. This happens in Column 3, so we know that 9 + 3 = 2 and that 3 is the solution to 2 - 9.

b.

Yes, it is possible to subtract any number from any other number in this system. This is true because each number occurs in every row and column exactly once, so we can always find a solution to a + x = b, no matter what numbers a and b are.

Answers will vary. For example, one pattern is that the first row and column are all zeros. Another is that the table is symmetrical about its main diagonal. Another is that some (but not all) rows have all 10 numbers.

b.

Answers will vary. Some patterns are pretty easy to explain -- every number multiplied by 0 is 0, so the row and column for 0 should be nothing but zeros. Some are much more difficult to explain, such as which rows will have all 10 numbers.

All the numbers with inverses are odd, while every even number has no inverse. The only exception to this pattern is 5; according to the multiplication table, any number multiplied by 5 will have a units digit of 0 or 5, so 1 is never a units digit. More explicitly, the numbers with inverses are relatively prime to 10 (they have no common factors, except 1, with 10). The numbers without inverses are not relatively prime to 10; they have common factors with 10 that are greater than 1.